Brahmagupta Got Division by Zero Wrong. The Tradition Argued About It for Five Hundred Years.

In chapter eighteen of the Brahmasphutasiddhanta, finished in 628 CE at Bhillamala in what is now western Rajasthan, Brahmagupta writes that zero divided by zero is zero. His is the earliest surviving book that sets out arithmetic rules for zero as a number in its own right rather than a mark holding an empty column, and this particular rule is false. Modern mathematics assigns 0/0 no value at all. Five centuries later Bhaskara II took the question up again, refused to let division by zero produce anything finite, named the result khahara, and described it as a quantity nothing added or removed can change. He was closer. He was not correct either, and a few lines away he says something plainly wrong. The sequence is the interesting part: not a tradition that got there first, but one that argued with itself in writing for five hundred years.

Fortunes, debts and the cipher

Chapter eighteen is the kuttakadhyaya, the chapter of the pulveriser, named for the algorithm for indeterminate linear equations that fills most of it, and treated by later Indian mathematicians as the algebra. Before Brahmagupta reaches the pulveriser he sets down the arithmetic of three sorts of quantity: dhana, a fortune, positive; rna, a debt, negative; and shunya, the cipher.

He does not treat the negatives as suspicious. A debt is an ordinary fact of commercial life in seventh-century western India, and the rules read like a merchant's bookkeeping generalised. The sum of two positives is positive, of two negatives negative; of a positive and a negative, their difference, and zero if they are equal. Zero subtracted from a negative leaves it negative. Then the sign law school algebra still teaches: the product of a negative and a positive is negative, of two negatives positive. Anything multiplied by zero is zero.

Worth measuring that against the alternative. Descartes, in 1637, still called the negative solutions of an equation fausses racines, false roots, keeping vraies, true, for the positive ones. Brahmagupta had no such difficulty, because he was never asking whether a debt exists. He was asking how debts combine, and that has an answer.

The rule that does not hold

Then the verse on division.

"A positive divided by a positive, or a negative by a negative, is positive. A zero divided by a zero is zero. A positive divided by a negative is negative; a negative divided by a positive is negative."

Brahmagupta, Brahmasphutasiddhanta, chapter eighteen, from the rules on positives, negatives and zero at verses 30 to 35

One caution on that citation. Editions do not number these verses identically: the division rule is 18.34 in the reckoning most often reproduced today and 18.35 in others, with the sign law one verse ahead of it in each. Check a printed text by content, not by decimal point.

The clauses about signs are correct. The clause about zero is not.

Division, defined properly, asks a question about multiplication: a divided by b is the number which, multiplied by b, gives back a. Ask that of 0/0 and there are too many answers. Zero times zero is zero, so 0 qualifies. Zero times seven is also zero, so 7 qualifies, and so does every other number. An operation that returns every possible value returns no value, and the modern refusal to define 0/0 is a refusal to pretend otherwise.

What makes the slip interesting is the harder case. For a positive or negative quantity divided by zero Brahmagupta gives no value, saying only that it is a quantity having zero as its divisor: he writes the fraction down and leaves it standing. He declines the case that looks impossible and answers the case that looks easy, and the easy-looking one trips him.

Why writing that in 628 was reasonable

The mistake is not carelessness. It is the natural end of the method, which is extension by pattern. Look at the run of rules he has just established. Zero added to a quantity leaves it unchanged, zero subtracted leaves it unchanged, zero multiplied by anything gives zero. Across an entire page, zero absorbs multiplication and stays neutral under addition. When the sequence reaches division and the numerator is also zero, "the answer is zero" completes the shape of everything above it.

What Brahmagupta lacked was not arithmetic ability. It was a habit mathematics acquired much later: checking whether a proposed answer is the only answer. Uniqueness as a requirement is not an obvious demand. It is the product of centuries of people finding out what goes wrong when you skip it.

There is a plainer reason it survived. These rules sit in the algebra chapter, where Brahmagupta is systematising the number line, not in the astronomical computation the book exists to carry out. A rule no calculation exercises is one nothing can falsify.

Two centuries on, it gets worse first

Around the middle of the ninth century, at the Rashtrakuta court under Amoghavarsha, the Jain mathematician Mahavira composed the Ganitasarasangraha. Where Brahmagupta had hedged, Mahavira takes a definite position: a number divided by zero remains unchanged.

That is a third answer, wrong in a new way. It makes division by zero the identity operation, so that seven divided by zero is seven, and it collides at once with a rule Mahavira himself keeps, that anything multiplied by zero is zero. Multiply both sides by zero and seven equals zero.

Three named mathematicians, in books that read and correct one another, give three different answers to the same question over five hundred years. Brahmagupta says zero, Mahavira says the number itself, Bhaskara says something unbounded. That is what a living mathematical tradition looks like from the inside. One that had been right from the beginning would have left no such trail, and would be much harder to believe.

Khahara, and the image Bhaskara reached for

Bhaskara II, born 1114 and dead around 1185, by tradition head of the astronomical observatory at Ujjain, addresses zero in both the Lilavati and the Bijaganita. He rejects the finite answers. A quantity divided by zero, he says, becomes a fraction whose denominator is zero, and that fraction gets its own name: khahara, from kha, one of the standard Sanskrit words for zero, which also means sky or empty space, and hara, divisor. He is not calling it a number like the others but a distinct kind of object, and in the Bijaganita he calls it an infinite quantity outright. The passages sit in the opening arithmetic of the Lilavati; printed editions number them variously in the middle forties, which is why they are described here rather than cited by number.

Then the part that has been quoted for eight hundred years. Having named it, he characterises it by behaviour, not by size.

"In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth."

Bhaskara II, in H. T. Colebrooke's 1817 English translation

Read past the theology and notice what is being specified. He is describing a fixed point of addition: an object x such that x plus any finite quantity is still x, and x minus any finite quantity is still x. That is a structural property rather than a magnitude, and it is the correct property to fix on. It is also, in twelfth-century Sanskrit, a property with no mathematical vocabulary and a very well developed theological one. The names of Vishnu in that vocabulary include Ananta, the endless, and Acyuta, the imperishable, and Colebrooke's "infinite and immutable God" renders that register. Which epithet stands in the verse is a question for whoever has the manuscript open.

This is not scripture doing mathematics, and it should not be sold as such. It is a mathematician borrowing a precise word from the only field around him that had already needed one.

The mistake Bhaskara made in the same breath

A verse or two earlier he lays down the rule for multiplication by zero, and it is unusually careful. Colebrooke's English has it as: the product of cipher is nought, but it must be retained as a multiple of cipher if any further operation impend. Multiply by zero and you get zero, but do not collapse it into a bare zero, because you may need it later.

Then the next rule. Cipher having become a multiplier, should nought afterwards become a divisor, the definite quantity must be understood to be unchanged. In modern notation: a quantity multiplied by zero and then divided by zero comes back as itself.

It does not. Once a number has been multiplied by zero the number is gone. Five times zero and nine times zero are the same object, and no rule applied afterwards can recover which you started with. It is the failure that defeated Brahmagupta arriving from the opposite direction: not too many answers this time, but a step that destroys the information the next step needs.

The rule is sometimes defended as an anticipation of limits, or of cancelling a common factor, and Bhaskara's own instruction to retain the multiple of cipher is the best evidence that defence has. If the zero is held as an unexpanded symbol, a tag on the quantity rather than a value substituted for it, striking it out again is legitimate bookkeeping and the rule works. If the zeros are numbers, it fails. Bhaskara does not mark the difference, and the honest description is that he did not have the distinction available to mark. Colebrooke's 1817 translation is out of copyright and on the Internet Archive, for anyone who would rather read the rules themselves.

What modern mathematics settles, and what it deliberately will not

Two separate things are going on, and popular accounts run them together.

First, in any field, meaning the rationals, the reals, the complex numbers, division by zero is undefined, and that is a structural necessity rather than an oversight. Suppose 1/0 were some number c. Then 1 equals c times 0. But anything times zero is zero, so 1 equals 0, and from there every number equals every other and arithmetic collapses. The prohibition is what keeps the rest standing.

Second, 0/0 is called an indeterminate form, which means something more specific than "undefined". In calculus you constantly meet ratios where numerator and denominator both shrink towards zero, and the ratio can approach anything depending on how they shrink. Take x divided by x as x goes to zero and the ratio is 1 throughout. Take x squared over x and it goes to 0. Take x over x cubed and it grows without bound. Same form, three destinations. The form carries no information, which is why it cannot be assigned a value.

Now the part that matters for Bhaskara. One standard modern structure does give a nonzero quantity divided by zero an answer: the projectively extended real line, and its complex cousin the Riemann sphere, which add a single unsigned point at infinity and define a/0 as that point for every a other than zero. There, infinity plus any finite number is still infinity, which is Bhaskara's invariance restated. Computers do a version of it billions of times a second: the IEEE 754 standard returns infinity for 1.0 divided by 0.0.

The same structure still refuses 0/0, and IEEE 754 returns NaN, not a number, for 0.0 divided by 0.0. The modern system that most resembles khahara declines the exact case Brahmagupta answered. More exotic algebras exist, the structures called wheels among them, in which division is total and every expression including 0/0 has a value, and the price is steep: in a wheel, x minus x is not always zero and x divided by x is not always one.

So nothing was settled. Brahmagupta gave a wrong answer, Mahavira a different wrong answer, Bhaskara a convention one modern structure also adopts, with a false rule attached. What happened between 628 and 1150 is that the question got sharper, a real achievement and a smaller one than the headline version.

The oldest zeros anyone can actually date

Behind the arithmetic sits a separate question with its own disputes: when does the written zero appear, and where.

The Gwalior evidence is the cleanest. At the Chaturbhuj temple in the Gwalior fort, a shrine to Vishnu carved out of the rock face, an inscription dated Vikrama Samvat 933, which is 876 CE, records a grant of land for a flower garden measuring 270 hastas by 187, and a daily gift of 50 garlands. The final digits of 270 and of 50 are small circles. It is dated in its own text, it is on stone, it is still in place, and it was long treated as India's oldest securely dated zero.

Then the Bakhshali manuscript: seventy fragmentary birch bark leaves of practical arithmetic, dug out of a field near Mardan in what is now Khyber Pakhtunkhwa in 1881, using a dot for zero. In 2017 the Bodleian Library radiocarbon dated three folios and announced results that would push the Indian zero back by centuries: 224 to 383 CE, 680 to 779, and 885 to 993.

Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk published a detailed objection in History of Science in South Asia volume 5 in 2017, pages 134 to 150. Their central point is codicological rather than chemical: a manuscript is dated by its latest datable folio, not its earliest, because a scribe can write on old material but cannot write on material harvested after his death. They also criticised the release of the findings by press statement and video ahead of peer review. Jan Houben added an objection from palaeography, pointing to Richard Salomon's dating of the Proto-Sharada script to no earlier than the mid seventh century. Those same scholars defend the manuscript on other grounds, arguing its zeros already work as numbers inside calculations rather than as placeholders.

Then the complication that ought to be stated plainly rather than left out. The oldest firmly dated decimal zero known anywhere is not in India. It is on a stele from Sambor on the Mekong, in Kratie province, Cambodia, catalogued as K-127, found in 1891 by the French official Adhemar Leclere and read by the epigrapher George Coedes, who published its significance in 1931. It carries a date in numerals, Shaka year 605, which converts to 683 CE, with a dot in the tens place. Some readings give Shaka 604 and 682 CE, an argument that turns on a single numeral and not on the zero.

That does not make Cambodia the birthplace of zero, and the inscription argues against the idea. It is dated in the Shaka era, an Indian calendar, and the numerals are Khmer forms of an Indian system that travelled east with Sanskrit and Indian administrative technique. By the 680s decimal place value with a zero was in routine bureaucratic use a long way from where it was developed, which puts Indian usage comfortably before the Gwalior stone. The Indian epigraphic record is thin, not late.

Why it is thin is visible in the stele's own history. K-127 went into storage at Angkor Conservation in Siem Reap as war closed in during the early 1970s, then vanished among thousands of unlabelled objects through the Khmer Rouge years, until the mathematician Amir Aczel tracked it to a shed on the site in 2013. He worked with the Cambodian ministry of culture to move it to the National Museum in Phnom Penh, where it went on display in January 2017. Aczel had died two years earlier.